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Symmetry of divisibility in elementary divisor rings. (Ukrainian. English summary) Zbl 1093.16025
Summary: K. Asano [J. Math. Soc. Japan 1, 73-78 (1949; Zbl 0037.30605), Osaka Math. J. 1, 52-61 (1949; Zbl 0041.16503)] and T. Nakayama [Proc. Imp. Acad. Jap. 16, 285-289 (1940; Zbl 0024.09904), ibid. 17, 53-56 (1941; Zbl 0026.05802)] proved for a semilocal ring the statement: if $$Ra=bR$$ is an ideal of the ring $$R$$, then $$Ra=aR=bR=Rb$$. In the paper this result is generalized for elementary divisor rings.
##### MSC:
 16U30 Divisibility, noncommutative UFDs 16D25 Ideals in associative algebras 16U80 Generalizations of commutativity (associative rings and algebras)
##### Keywords:
elementary divisor rings; duo-rings; duo-elements